论文信息 - Concentration of points on curves in finite fields

Concentration of points on curves in finite fields

We obtain analogues of several recent bounds on the number of solutions of polynomial congruences modulo a prime with variables in short intervals in the case of polynomial equations in high degree extensions of finite fields. In these settings low-dimensional affine spaces play the role of short intervals and thus several new ideas are required.

Igor E. Shparlinski | Javier Cilleruelo | I. Shparlinski | J. Cilleruelo

[1] Philippe Piret. On the number of divisors of a polynomial over GF(2) , 1984, AAECC.

[2] Todd Cochrane,et al. The congruence x1x2≡x3x4(modm) and mean values of character sums , 2010 .

[3] Igor E. Shparlinski,et al. Isomorphism classes of elliptic curves over a finite field in some thin families , 2012 .

[4] Enrico Bombieri,et al. The number of integral points on arcs and ovals , 1989 .

[5] Todd Cochrane,et al. The Congruencex1x2≡x3x4(modp), the Equationx1x2≡x3x4, and Mean Values of Character Sums , 1996 .

[6] M. Garaev,et al. The equation x1x2=x3x4+λ in fields of prime order and applications , 2008 .

[7] Igor E. Shparlinski,et al. On the concentration of points of polynomial maps and applications , 2012 .

[8] Igor E. Shparlinski,et al. Points on curves in small boxes en applications , 2011, 1111.1543.

[9] Trevor D. Wooley,et al. Vinogradov's mean value theorem via efficient congruencing, II , 2011, 1112.0358.

[10] Igor Shparlinski,et al. On congruences with products of variables from short intervals and applications , 2012, 1203.0017.

[11] M. Z. Garaev,et al. Concentration of Points on Two and Three Dimensional Modular Hyperbolas and Applications , 2010, 1007.1526.

[12] Igor E. Shparlinski,et al. On the Hidden Shifted Power Problem , 2011, SIAM J. Comput..

[13] Trevor D. Wooley,et al. Vinogradov's mean value theorem via efficient congruencing, II , 2011, 1112.0358.